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Identify the operator norm of a function on $\ell^2(\mathbb{N})$

Suppose that I know that $$T(x_n)=x_{n+1}$$ and $$||T||=\max_{n\in \mathbb{N}} \sqrt{n+1}$$
I’ve identified that this is an isometry, so we know that $$||T(x_n)||=\sqrt{n+1} \quad \forall n\in \mathbb{N}$$
What I need to show is that the norm is given by $\frac{1}{2}$.
I’ve tried to construct some $x$ that is in the range of the transformation and in this case $||T||$ is determined by the value of $x$, I mean, if $x=e_n$ then the $||T||=n+1$ but that’s no good for the calculation.
I’m not sure about if I can construct a sequence of elements that are easy to identify $||T||$.
I need to show that $||T||\le \frac{1}{2}$ is it possible?

A:

This is a well-known fact. Consider the set $\{ n \in \mathbb N : n \le N\}$. We have
$$\begin{array}{cccccccc} ||T|| & = & \max \{ \sqrt{k+1} : k \le N\} & = & \sqrt{N+1} \end{array}$$
Now, for any $k$, if $n \in \{ 1, \ldots, N\}$, then
k \le N \implies (k+1)^2 \

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